复旦大学：《投资学讲义》（英文版） Chepter 7 Arbitrage Pricing Theor

INVESTMENTS Chapter 7 arbitrage Pricing Theory

INVESTMENTS Arbitrage Arbitrage Pricing Theory Pricing Theory Chapter 7 Chapter 7

INVESTMENTS Arbitrage pricing theor Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit Since no investment is required, an investor can create large positions to secure large levels of profit In efficient markets, profitable arbitrage opportunities will quickly disappear

INVESTMENTS Arbitrage Pricing Theory Arbitrage Pricing Theory Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit „ Since no investment is required, an investor can create large positions to secure large levels of profit „ In efficient markets, profitable arbitrage opportunities will quickly disappear

INVESTMENTS Arbitrage Example from Text Current Expected Standard Stock PriceS Return 0 evo 10 25.0 2958 ABCD 20.0 33.91 32.5 48.15 22.5 8.58

INVESTMENTS Current Expected Standard Stock Price$ Return% Dev.% A 10 25.0 29.58 B 10 20.0 33.91 C 10 32.5 48.15 D 10 22.5 8.58 Arbitrage Example from Text Arbitrage Example from Text pp. 308 pp. 308 -310

INVESTMENTS Arbitrage portfolio ean S D. Correlation Portfolio A.B.C25.83 094 22.25 8.58

INVESTMENTS Mean S.D. Correlation Portfolio A,B,C 25.83 6.40 0.94 D 22.25 8.58 Arbitrage Portfolio Arbitrage Portfolio

INVESTMENTS Arbitrage action and retl E Ret D St. dev Short 3 shares of d and buy l of a, b& c to form p You earn a higher rate on the investment than you pay on the short sale

INVESTMENTS Arbitrage Action and Returns Arbitrage Action and Returns E. Ret. St.Dev. * P * D Short 3 shares of D and buy 1 of A, B & C to form P You earn a higher rate on the investment than you pay on the short sale

INVESTMENTS Factor model of asset returns Suppose that asset returns are driven by a few common factors and diversifiable noise: r=E(r)+6,.+ +u Whe Er is the expected return on asset i f f,f are news on common factors driving all asset returns. f=F-E(F) d b gives how sensitive the return on asset i with respect to news on the k-th factor--is called the loading of asset i on factor fi ui is the idiosyncratic component in asset i's return that is unrelated to other asset returns f, Ji, x, u, have zero means

INVESTMENTS Factor model of asset returns Factor model of asset returns „ Suppose that asset returns are driven by a few common factors and diversifiable noise: „ Where „ is the expected return on asset i; „ are news on common factors driving all asset returns. „ gives how sensitive the return on asset i with respect to news on the k-th factor—is called the loading of asset i on factor „ is the idiosyncratic component in asset i’s return that is unrelated to other asset returns „ have zero means. i i i iK K ui r = E r + b f + b f + ~ ... ~ ( ) 1 1 Eri K f f f ~ ,..., ~ , ~ 1 2 ( ) ~ k Fk E Fk f = − bik k f ~ u i ~ K ui f f f , ~ ,..., ~ , ~ 1 2

INVESTMENTS exanple Common factors driving asset returns may include gnp interest rates, inflation, etc. Let fit be the news on interest rate. Before a board meeting of the fed, the market expect the Fed not to change the interest rate. After the meeting, Greenspan announces that a There is no change in interest rate---no news"' fint=0 a There is a 4% increase in interest rate--positive surprise 0.25% What should be the sign of the factor loadings on f be for fixed income securities, stocks, commodity futures?

INVESTMENTS example example „ Common factors driving asset returns may include GNP, interest rates, inflation, etc. Let be the news on interest rate. Before a board meeting of the Fed, the market expect the Fed not to change the interest rate. After the meeting, Greenspan announces that: „ There is no change in interest rate---”no news” „ There is a ¼% increase in interest rate—positive surprise „ What should be the sign of the factor loadings on , be for fixed income securities, stocks, commodity futures? int ~ f 0 ~ fint = 0.25 % ~ fint = int ~ f

INVESTMENTS Properties of factor model The following results provide the building blocks of apt 1. Any diversified portfolio p is exposed only to factor risks E(n)+bn+…b KJK

INVESTMENTS Properties of factor models Properties of factor models „ The following results provide the building blocks of APT. „ 1. Any diversified portfolio p is exposed only to factor risks p p p pK K r E r b f b f ~ ... ~ ( ) = + 1 1 +

INVESTMENTS corrIne u a diversified portfolio, that is not exposed to any factor b=b box =0 ) must offer risk-free rat There al ways exists portfolios that are exposed only to the risk of a single factor. pk=Ipx+bpe k Example: suppose two well diversified portfolios, both exposed only to the risk of the first two factors f, f 1=0.2+f1+0.52,2=0.3+2f+15

INVESTMENTS continued continued „ A diversified portfolio, that is not exposed to any factor risk( ), must offer risk-free rate; „ There always exists portfolios that are exposed only to the risk of a single factor. „ Example: suppose two well diversified portfolios, both exposed only to the risk of the first two factors 2 ... 0 1 b p = b p = = bpK = pk pk pk k r r b f ~ = + 1 2 ~ , ~ f f , ~ 0.5 ~ 1 0.2 1 2 r = + f + f , ~ 1.5 ~ 2 0.3 2 1 2 r = + f + f

INVESTMENTS continued I A portfolio Pk, that has unitary risk of factor k, offer a expected return with the factor risk pk fk such a portfolio is called a factor portfolio for factor k, and A-s is the premium of factor k 1 Example. In the above example, we found portfolio p that bears only the risk of factor 1. Its loading is 0.5. Consider following portfolio pi 1 200% invested in p and% invested in the risk-free portfolio PO

INVESTMENTS continued continued „ A portfolio P k, that has unitary risk of factor k, offer a expected return with the factor risk: „ „ such a portfolio is called a factor portfolio for factor k, and „ is the premium of factor k „ Example. In the above example, we found portfolio p that bears only the risk of factor 1. Its loading is 0.5. Consider following portfolio p1: „ 200% invested in p and –100% invested in the risk-free portfolio P0. pk fk r = r fk f r − r